Mirror quintics, discrete symmetries and Shioda maps

In a recent paper, Doran, Greene and Judes considered one parameter families of quintic threefolds with finite symmetry groups. A surprising result was that each of these six families has the same Picard–Fuchs equation associated to the holomorphic $3$-form. In this paper we give an easy argument, involving the family of Mirror Quintics, which implies this result. Using a construction due to Shioda, we also relate certain quotients of these one-parameter families to the family of Mirror Quintics. Our constructions generalize to degree $n$ Calabi–Yau varieties in $(n-1)$-dimensional projective space.